Are you thinking about rationalizing the denominator?

Relevant wiki: Telescoping Series - Product

The sum can be written as:

$\begin{aligned}\mathcal K=&\displaystyle\sum_{r=1}^{24}\dfrac{1}{(r+1)\sqrt r+r\sqrt{r+1}}\\=&\displaystyle\sum_{r=1}^{24}\dfrac1{\sqrt{r(r+1)}(\sqrt{r+1}+\sqrt r)}\\&\text{Rationalising the denominator:-}\\=&\displaystyle\sum_{r=1}^{24}\dfrac{\sqrt{r+1}-\sqrt r}{\sqrt{r(r+1)}}\\=&\displaystyle\sum_{r=1}^{24}\dfrac1{\sqrt r}-\dfrac1{\sqrt{r+1}}\end{aligned}$

$(\mathbf{\color{#0C6AC7}{A~Telescopic~Series}})$

$\therefore\mathcal K=\dfrac1{\sqrt 1}-\dfrac1{\sqrt{25}}=\dfrac 45=\boxed{0.80}$